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Question

Which of the the following Expressions,are cyclic polynomials.


A
Both 1 and 2
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B
Only 2
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C
Only 4
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D
Both 1, 2 and 4
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Solution

The correct option is C Both 1, 2 and 4
A polynomial is said to be cyclic if f(x,y,z)=f(y,z,x)
(1)x+y+z
We have f(x,y,z)=x+y+z
f(y,z,x)=y+z+x=x+y+z=f(x,y,z)
Therefore (1) is cyclic.
(2)(xy)2+(yz)2+(zx)2
We have f(x,y,z)=(xy)2+(yz)2+(zx)2
f(y,z,x)=(yz)2+(zx)2+(xy)2=f(x,y,z)=(xy)2+(yz)2+(zx)2=f(x,y,z)
Therefore (2) is cyclic.
(3)x(y+z)y(z+x)+z(x+y)
We have f(x,y,z)=x(y+z)y(z+x)+z(x+y)
f(y,z,x)=y(z+x)z(x+y)+x(y+z)f(x,y,z)
Therefore (3) is not cyclic.
(4)x2+y2+z2
We have f(x,y,z)=x2+y2+z2
f(y,z,x)=y2+z2+x2=x2+y2+z2=f(x,y,z)
Therefore (4) is cyclic.
Hence (1),(2),(4) are cyclic.

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